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Connectivity of aMRI and fMRI data
Keith WorsleyArnaud CharilJason Lerch
Francesco Tomaiuolo
Department of Mathematics and Statistics, McConnell Brain Imaging Centre, Montreal Neurological Institute,
McGill University
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Effective connectivity• Measured by the correlation between residuals at
pairs of voxels:
Voxel 2
Voxel 1
++ +++ +
Activation onlyVoxel 2
Voxel 1++
+
+
+
+
Correlation only
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Types of connectivity
• Focal
• Extensive
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cor=0.58
Focal correlation
n = 120frames
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Extensive correlation
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Methods
1. Seed
2. Iterated seed
3. Thresholding correlations
4. PCA
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Method 1: ‘Seed’
Friston et al. (19??): Pick one voxel, then find all others that are correlated with it:
Problem: how to pick the ‘seed’ voxel?
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Method 2: Iterated ‘seed’
• Problem: how to find the rest of the connectivity network?
• Hampson et al., (2002): Find significant correlations, use them as new seeds, iterate.
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Method 3: All correlations
• Problem: how to find isolated parts of the connectivity network?
• Cao & Worsley (1998): find all correlations (!)
• 6D data, need higher threshold to compensate
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Thresholds are not as high as you might think:
E.g. 1000cc search region, 10mm smoothing, 100 df, P=0.05:
dimensions D1 D2 Cor T
Voxel1 - Voxel2 0 0 0.165 1.66
One seed voxel - volume 0 3 0.448 4.99
Volume – volume (auto-correlation) 3 3 0.609 7.64
Volume1 – volume2 (cross-correlation) 3 3 0.617 7.81
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Practical details
• Find threshold first, then keep only correlations > threshold
• Then keep only local maxima i.e.cor(voxel1, voxel2)
> cor(voxel1, 6 neighbours of voxel2),
> cor(6 neighbours of voxel1, voxel2),
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Method 4: Principal Components Analysis (PCA)
• Friston et al: (1991): find spatial and temporal components that capture as much as possible of the variability of the data.
• Singular Value Decomposition of time x space matrix:
Y = U D V’ (U’U = I, V’V = I, D = diag)
• Regions with high score on a spatial component (column of V) are correlated or ‘connected’
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Which is better:
thresholding correlations,
or
PCA?
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Summary
Extensive correlationFocal correlation
Thresholding T statistic
(=correlations)
PCA
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1000First scan of fMRI data
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T statistic for hot - warm effect
0 100 200 300
870880890 hot
restwarm
Highly significant effect, T=6.59
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800
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No significant effect, T=-0.74
0 100 200 300
790800810
Drift
Time, seconds
fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …
T = (hot – warm effect) / S.d. ~ t110 if no effect
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Temporal components (sd, % variance explained)
0.68, 46.9%
0.29, 8.6%
0.17, 2.9%
0.15, 2.4%
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Spatial components
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PCA of time space:
1: excludefirst frames
2: drift
3: long-range correlationor anatomicaleffect: removeby converting to % of brain
4: signal?
Frame
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MS lesions and cortical thickness(Arnaud et al., 2004)
• n = 425 mild MS patients
• Lesion density, smoothed 10mm
• Cortical thickness, smoothed 20mm
• Find connectivity i.e. find voxels in 3D, nodes in 2D with high
cor(lesion density, cortical thickness)
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0 10 20 30 40 50 60 70 801.5
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Average lesion volume
Ave
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n=425 subjects, correlation = -0.56826
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Normalization
• Simple correlation:
Cor( LD, CT )
• Subtracting global mean thickness:
Cor( LD, CT – avsurf(CT) )
• And removing overall lesion effect:
Cor( LD – avWM(LD), CT – avsurf(CT) )
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distance (mm)
corr
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Same hemisphere
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Correlation = 0.091943
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Correlation = -0.1257
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threshold
thresholdthreshold
threshold
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Deformation Based Morphometry (DBM) (Tomaiuolo et al., 2004)
• n1 = 19 non-missile brain trauma patients, 3-14 days in coma,
• n2 = 17 age and gender matched controls
• Data: non-linear vector deformations needed to warp each MRI to an atlas standard
• Locate damage: find regions where deformations are different, hence shape change
• Is damage connected? Find pairs of regions with high canonical correlation.
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T = sqrt(df) cor / sqrt (1 - cor2)
T max = 7.81P=0.00000004
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PCA, component 1
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T max = 4.17P = 0.59
T, extensive correlation
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PCA, focal correlation
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Modulated connectivity
• Looking for correlations not very interesting – ‘resting state networks’
• More intersting: how does connectivity change with- task or condition (external)- response at another voxel (internal)
• Friston et al., (1995): add interaction to the linear model:
Data ~ task + seed + task*seed Data ~ seed1 + seed2 + seed1*seed2
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Fit a linear model for fMRI time series with AR(p) errors
• Linear model: ? ? Yt = (stimulust * HRF) b + driftt c + errort
• AR(p) errors: ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt
• Subtract linear model to get residuals.• Look for connectivity.
unknown parameters